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bonufazy [111]
3 years ago
7

Determine whether or not f is a conservative vector field. if it is, find a function f such that f = ∇f. if it is not, enter non

e. f(x, y) = (2x − 4y) i + (−4x + 10y − 9) j
Mathematics
1 answer:
mixer [17]3 years ago
6 0
\mathbf f(x,y)=(2x-4y)\,\mathbf i+(-4x+10y-9)\,\mathbf j

is conservative if

\dfrac{\partial(2x-4y)}{\partial y}=\dfrac{\partial(-4x+10y-9)}{\partial x}

It's clear that this is true, since both partial derivatives reduce to -4, so \mathbf f is indeed a conservative vector field.

Since \mathbf f(x,y) is conservative, there must be some scalar function f(x,y) such that

\nabla f(x,y)=\dfrac{\partial f}{\partial x}\,\mathbf i+\dfrac{\partial f}{\partial y}\,\mathbf j=\mathbf f(x,y)
\implies\dfrac{\partial f}{\partial x}=2x-4y

Integrating with respect to x yields

\displaystyle\int\frac{\partial f}{\partial x}\,\mathrm dx=f(x,y)=x^2-4xy+g(y)

and differentiating with respect to y yields

\dfrac{\partial f}{\partial y}=-4x+g'(y)=-4x+10y-9
\implies g'(y)=10y-9
\implies g(y)=\displaystyle\int g'(y)\,\mathrm dy=5y^2-9y+C

Thus the scalar potential of \mathbf f(x,y) is

f(x,y)=x^2-4xy+5y^2-9y+C

where C is an arbitrary constant.
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